What is the integral from pi to 2pi of sin(x/2) ?

The integral from pi to 2pi of sin(x/2) is 2

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integral from pi to 2pi of sin(x/2)

\int_{π}^{2 π} sin (\frac{x}{2}) d x

2

Solution steps

\int_{π}^{2 π} sin (\frac{x}{2}) d x

Apply u-substitution

: \int_{\frac{π}{2}}^{π} sin (u) \cdot 2 d u

= \int_{\frac{π}{2}}^{π} sin (u) \cdot 2 d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{\frac{π}{2}}^{π} sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= 2 [- cos (u)]_{\frac{π}{2}}^{π}

Compute the boundaries:

1

= 2 \cdot 1

Simplify

= 2

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What is the integral from pi to 2pi of sin(x/2) ?
The integral from pi to 2pi of sin(x/2) is 2